Find the distance between the point ${(2, -4)}$ and the line $\enspace {y = \dfrac{1}{3}x + 2}\thinspace$. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$
First, find the equation of the perpendicular line that passes through ${(2, -4)}$ The slope of the blue line is ${\dfrac{1}{3}}$ , and its negative reciprocal is ${-3}$ Thus, the equation of our perpendicular line will be of the form $\enspace {y = -3x + b}\thinspace$ We can plug our point, ${(2, -4)}$ , into this equation to solve for ${b}$ , the y-intercept. $-4 = {-3}(2) + {b}$ $-4 = -6 + {b}$ $-4 + 6 = {b} = 2$ The equation of the perpendicular line is $\enspace {y = -3x + 2}\thinspace$ We can see from the graph (or by setting the equations equal to one another) that the two lines intersect at the point ${(0, 2)}$ . Thus, the distance we're looking for is the distance between the two red points. The distance formula tells us that the distance between two points is equal to: $\sqrt{( x_{1} - x_{2} )^2 + ( y_{1} - y_{2} )^2}$ Plugging in our points ${(2, -4)}$ and ${(0, 2)}$ gives us: $\sqrt{( {2} - {0} )^2 + ( {-4} - {2} )^2}$ $= \sqrt{( 2 )^2 + ( -6 )^2} = \sqrt{40} = 2\sqrt{10}$ The distance between the point ${(2, -4)}$ and the line $\thinspace {y = \dfrac{1}{3}x + 2}\enspace$ is $\thinspace2\sqrt{10}$.